Realm Of Parameterized Complexity Research Articles

A single exponential-time FPT algorithm for cactus contraction

For a collection F of graphs, the F-Contraction problem takes a graph G and an integer k as input and decides if G can be modified to some graph in F using at most k edge contractions. The F-Contraction problem is NP-Complete for several graph classes F. Heggernes et al. (2014) [4] initiated the study of F-Contraction in the realm of parameterized complexity. They showed that it is FPT if F is the set of all trees or the set of all paths. In this paper, we study F-Contraction where F is the set of all cactus graphs and show that we can solve it in 2O(k)⋅|V(G)|O(1) time.

On the Parameterized Complexity of Maximum Degree Contraction Problem

In the Maximum Degree Contraction problem, the input is a graph G on n vertices, and integers k, d, and the objective is to check whether G can be transformed into a graph of maximum degree at most d, using at most k edge contractions. A simple brute-force algorithm that checks all possible sets of edges for a solution runs in time $$n^{\mathcal {O}(k)}$$ . As our first result, we prove that this algorithm is asymptotically optimal, upto constants in the exponents, under Exponential Time Hypothesis (ETH). Belmonte, Golovach, van’t Hof, and Paulusma studied the problem in the realm of parameterized complexity and proved, among other things, that it admits an FPT algorithm running in time $$(d + k)^{2k} \cdot n^{\mathcal {O}(1)} = 2^{\mathcal {O}(k \log (k+d) )} \cdot n^{\mathcal {O}(1)}$$ , and remains NP-hard for every constant $$d \ge 2$$ (Acta Informatica (2014)). We present a different FPT algorithm that runs in time $$2^{\mathcal {O}(dk)} \cdot n^{\mathcal {O}(1)}$$ . In particular, our algorithm runs in time $$2^{\mathcal {O}(k)} \cdot n^{\mathcal {O}(1)}$$ , for every fixed d. In the same article, the authors asked whether the problem admits a polynomial kernel, when parameterized by $$k + d$$ . We answer this question in the negative and prove that it does not admit a polynomial compression unless $$\textsf {NP}\subseteq \textsf {coNP}/poly$$ .

Gehrlein stability in committee selection: parameterized hardness and algorithms

In a multiwinner election based on the Condorcet criterion, we are given a set of candidates, and a set of voters with strict preference ranking over the candidates. A committee is \em weakly Gehrlein stable (WGS) if each committee member is preferred to each non-member by at least half of the voters. Recently, Aziz et al. [IJCAI 2017] studied the computational complexity of finding a WGS committee of size k. They show that this problem is \sf NP -hard in general and polynomial time solvable when the number of voters is odd. In this article, we initiate a systematic study of the problem in the realm of parameterized complexity. We first show that the problem is \sf W [1]-hard when parameterized by the size of the committee. To overcome this intractability result, we use a known reformulation of WGS as a problem on directed graphs and then use parameters that measure the "structure'' of these directed graphs. In particular, we consider the majority graph, defined as follows: there is a vertex corresponding to each candidate, and there is a directed arc from a candidate c to $c'$ if the number of voters that \em prefer c over $c'$ is more than those that prefer $c'$ over c. The problem of finding WGS committee of size k corresponds to finding a vertex subset X of size k in the majority graph with the following property: the set X contains no vertex outside the committee that has an in-neighbor in X. Observe that the polynomial time algorithm of Aziz et al. [IJCAI 2017] corresponds to solving the problem on a tournament (a complete graph with orientation on edges). Thus, natural parameters to study our problem are "closeness'' to being a tournament. We define closeness as the number of missing arcs in the given directed graph and the number of vertices we need to delete from the given directed graph such that the resulting graph is a tournament. We show that the problem is fixed parameter tractable (\sf FPT ) and admits linear kernels with respect to closeness parameters. Finally, we also design an exact exponential time algorithm running in time $OO(1.2207^nn^OO(1) )$. Here, n denotes the number of candidates.

Generalized Pseudoforest Deletion: Algorithms and Uniform Kernel

Feedback Vertex Set (FVS) is one of the most well-studied problems in the realm of parameterized complexity. In this problem we are given a graph $G$ and a positive integer $k$ and the objective is to test whether there exists $S\subseteq V(G)$ of size at most $k$ such that $G-S$ is a forest. Thus, FVS is about deleting as few vertices as possible to get a forest. The main goal of this paper is to study the following interesting problem: How can we generalize the family of forests such that the nice structural properties of forests and the interesting algorithmic properties of FVS can be extended to problems on this class? Toward this we define a graph class, ${\cal F}_l$, that contains all graphs where each connected component can be transformed into a forest by deleting at most $l$ edges. A graph in the class ${\cal F}_1$ is known as pseudoforest in the literature and we call a graph in ${\cal F}_l$ an $l$-pseudoforest. We study the problem of deleting $k$ vertices to get into ${\cal F}_l$, l-pseudoforest Deletion, in the realm of parameterized complexity. We show that l-pseudoforest Deletion admits an algorithm with running time $c_l^k n^{\mathcal{O}(1)}$ and admits a kernel of size $f(l)k^2$. Thus, for every fixed $l$ we have a kernel of size $\mathcal{O}(k^2)$. That is, we get a uniform polynomial kernel for l-pseudoforest Deletion. Our algorithms and uniform kernels involve the use of the expansion lemma and protrusion machinery.

Parameterized Algorithms for Non-separating Trees and Branchings in Digraphs

A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer $$\ell $$l, we can test whether D contains $$\ell $$l arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc-Disjoint Branchings and Non-Disconnecting Out-Branching. In Arc-Disjoint Branchings (Non-Disconnecting Out-Branching), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems. Non-Disconnecting Out-Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel. Arc-Disjoint Branchings is FPT on strong digraphs. The algorithm for Non-Disconnecting Out-Branching runs in time $$2^{\mathcal {O}(k)}n^{\mathcal {O}(1)}$$2O(k)nO(1) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.

A Polynomial Kernel for Feedback Arc Set on Bipartite Tournaments

In the k-Feedback Arc/Vertex Set problem we are given a directed graph D and a positive integer k and the objective is to check whether it is possible to delete at most k arcs/vertices from D to make it acyclic. Dom et al. (J. Discrete Algorithm 8(1):76---86, 2010) initiated a study of the Feedback Arc Set problem on bipartite tournaments (k-FASBT) in the realm of parameterized complexity. They showed that k-FASBT can be solved in time O(3.373 k n 6) on bipartite tournaments having n vertices. However, until now there was no known polynomial sized problem kernel for k-FASBT. In this paper we obtain a cubic vertex kernel for k-FASBT. This completes the kernelization picture for the Feedback Arc/Vertex Set problem on tournaments and bipartite tournaments, as for all other problems polynomial kernels were known before. We obtain our kernel using a non-trivial application of "independent modules" which could be of independent interest.

Implicit branching and parameterized partial cover problems

Covering problems are fundamental classical problems in optimization, computer science and complexity theory. Typically an input to these problems is a family of sets over a finite universe and the goal is to cover the elements of the universe with as few sets of the family as possible. The variations of covering problems include well-known problems like Set Cover, Vertex Cover, Dominating Set and Facility Location to name a few. Recently there has been a lot of study on partial covering problems, a natural generalization of covering problems. Here, the goal is not to cover all the elements but to cover the specified number of elements with the minimum number of sets. In this paper we study partial covering problems in graphs in the realm of parameterized complexity. Classical (non-partial) version of all these problems has been intensively studied in planar graphs and in graphs excluding a fixed graph H as a minor. However, the techniques developed for parameterized version of non-partial covering problems cannot be applied directly to their partial counterparts. The approach we use, to show that various partial covering problems are fixed parameter tractable on planar graphs, graphs of bounded local treewidth and graph excluding some graph as a minor, is quite different from previously known techniques. The main idea behind our approach is the concept of implicit branching. We find implicit branching technique to be interesting on its own and believe that it can be used for some other problems.

Betweenness parameterized above tight lower bound

We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the Betweenness problem parameterized above its tight lower bound, which is stated as follows. For a set V of variables and set C of constraints “ v i is between v j and v k ”, decide whether there is a bijection from V to the set { 1 , … , | V | } satisfying at least | C | / 3 + κ of the constraints in C . Our result solves an open problem attributed to Benny Chor in Niedermeier's monograph “Invitation to Fixed-Parameter Algorithms”. The betweenness problem is of interest in molecular biology. An approach developed in this paper can be used to determine parameterized complexity of a number of other optimization problems on permutations parameterized above or below tight bounds.